lecture8_web - L8-1EEL 6550 Error-control Coding Lecture...

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Unformatted text preview: L8-1EEL 6550 Error-control Coding Lecture 8FieldsDEFNAfieldis a commutative ring with identity in which every element has aninverse under.Essentially, a field is:*a set of elementsF*with two binary operations+(addition)and(multiplication)*+,, andinversescan be used to doaddition, subtraction, multiplication, anddivisionwithout leaving the setDEFNFormal definition:Afieldconsists of a setFand two binary operations+andthat satisfy the following properties:1.Fforms acommutative group under addition (+).The additive identity is labeled 0.2.F- {}forms acommutative group under multiplication().Themultiplicative identity is labeled 1.3. The operation.distributes over+:a(b+c) = (ab) + (ac).Examples of Infinite Fields*The rational numbers*The integers do not form a field because they do not form a group under . (Thereare no multiplicative inverses.)*The real numbers*The complex numbersFinite FieldsFinite fieldsare more commonly known asGalois Fieldsafter their discovererAGalois fieldwithpmembers is denotedGF(p)Every field must have at least 2 elements:*the additive identity 0, and*the multiplicative identity 1L8-2There exists a finite field with 2 elements: thebinary field, denotedGF(2)F={,1}+defined as modulo-2 addition+1111defined as modulo-2 multiplication111It is easy to verify thatdistributes of+by trying each of the 8 possible combinationsGiven a prime numberp, the integers{,1,2, . . . , p-1}form a field under modulopaddition and multiplication....
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This note was uploaded on 03/20/2012 for the course EEL 6650 taught by Professor Shea during the Spring '12 term at University of Florida.

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lecture8_web - L8-1EEL 6550 Error-control Coding Lecture...

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